Does rotation by exact angle become low-precise in blender?
It seems you have made a slight mathematical error.
The plane has a length of 2 (by default), so if you look at the diagram, the hypotenuse of one of the smaller right triangles is half the length of the plane, so it is $\frac{2}{2} = 1$ unit long.  Now, $\cos(60) = \frac{1}{2}$ so the shorter leg will have length $(1)\cos(60) = (1)\cdot(\frac{1}{2}) = \frac{1}{2}$.  Here is where you made a mistake: $\sin(60) = \frac{\sqrt{3}}{2}$, so the longer leg will have length $(1)\cdot\sin(60) = \frac{\sqrt{3}}{2}$, not $2$.
Since I stink at making mathematical diagrams on paint and that is all I have on my tablet, here is a diagram I drew: